Limit cycles for a generalization of polynomial Liénard differential systems

Abstract

We study the number of limit cycles of the polynomial differential systems of the forṁ x = y − f 1 (x)y,ẏ = −x − g 2 (x) − f 2 (x)y, where f 1 (x) = εf 11 (x) + ε 2 f 12 (x) + ε 3 f 13 (x), g 2 (x) = εg 21 (x) + ε 2 g 22 (x) + ε 3 g 23 (x) and f 2 (x) = εf 21 (x)+ε 2 f 22 (x)+ε 3 f 23 (x) where f 1i , f 2i and g 2i have degree l, n and m respectively for each i = 1, 2, 3, and ε is a small parameter. Note that when f 1 (x) = 0 we obtain the generalized polynomial Liénard differential systems. We provide an accurate upper bound of the maximum number of limit cycles that this differential system can have bifurcating from the periodic orbits of the linear centerẋ = y,ẏ = −x using the averaging theory of third order.

Uniqueness and Non-uniqueness of Limit Cycles for Piecewise Linear Differential Systems with Three Zones and No Symmetry

“…For smooth Liénard systems there are many results on the non-existence, existence and uniqueness of limit cycles, see for instance [1,6,8,14,21,24,29,34,36]. Going beyond the smooth case a natural step is to allow non-smoothness while keeping the continuity, as has been done in some recent works [10,17,18,25].…”

Two Limit Cycles in Liénard Piecewise Linear Differential Systems

“…Due to the fact that this Hilbert problem becomes up to now intractable (see [16,18]), Smale in [30] proposed to study this problem restricting it to polynomial Liénard differential systems. In the case of smooth Liénard systems there are many results on the non-existence, existence and uniqueness of limit cycles, see for instance [1,4,6,12,17,20,25,31,33]. Going beyond the smooth case a natural step is to allow non-smoothness while keeping the continuity, as it has been done in some previous works [9,14,15,21,24].…”

On the limit cycles of a class of Kukles type differential systems

“…We are particularly interested in study the maximum number of small amplitude limit cycles of a class of systems as in (1) which can coexist with closed invariant algebraic curves [6,5,24,23,28]. In this way, our methods are strongly influenced by the methods and ideas from [14,15,16], and following [13] we apply the averaging theory in order to study the maximum number of limit cycles which can bifurcate from the linear centerẋ = −y,ẏ = x, perturbed in a special class of systems. Specifically, we consider the following system in the plane:ẋ = −y,…”